How aggressive a driver is? - A quantitative analysis

Consider a bottleneck in a road through which only one car can pass through. Suppose that at a time the car passing will have the most aggressive driver in queue and that the aggressiveness of an individual is measured by an attribute $A \equiv N \tau^{\sigma}$ where the quantity $N$ varies randomly from person to person in the range 0 to 1, $\tau$ is the time for which the driver is waiting in the bottleneck and the parameter $\sigma$ is the same for all individuals. Thus, we assume that the aggressiveness depends on the nature of the individual and increases with waiting time in a traffic jam. In support of the algebraic form of $A$, we show (numerically and analytically) that our hypothesis implies that the probability of waiting for a time $\tau$ will be $P(\tau) \propto \tau^{\alpha}$ with the value of $\alpha$ fixed by $\sigma$. Empirical studies confirm such variation in $P(\tau)$ with an exponent of 3.0 to 3.5 in two different cities of India and 1.5 for a traffic intersection in Germany. There is a possibility that the parameter $\sigma$ (and hence $\alpha$) is characteristic of a geographical region.